Abstract
We lift indicator-style functions—sgn, Heaviside H, ramp/ReLU, and clip—to symbolic numerals (m, a). Each operator selects among branches; we make branch choice on the magnitude and carry the alignment from the selected source (input or constant), with explicit and reproducible tie policies. Under collapse phi(m,a) = m, all formulas reduce to their classical counterparts.
Alignment policy for selectors (default)
When a piecewise function selects a branch:
result = selected_argument
alignment(result) = alignment(selected_argument)
- If a constant is selected (e.g.,
0or1), its default alignment is+1unless you explicitly tag it. - This “carry-from-source” rule tracks whether the output came from a saturated constant (stable) or from the input (inherits input stability).
Tie-breaking at thresholds (manifest knobs).
At points where branches meet (e.g., m = 0), choose one of:
selector_tie = "prefer_constant" # default for clamp/clip boundaries
selector_tie = "prefer_input" # keep input branch on ties
selector_tie = "alignment_sign" # choose upper branch if a >= 0 else lower
For Heaviside at 0, you may also choose:
heaviside_zero = "half" | "right" | "left" | "alignment_sign"
Document choices on the page.
Sign function (sgn)
Classical:
sgn(m) = -1 if m < 0
0 if m = 0
+1 if m > 0
Symbolic lift (selector-carry):
sgn*(m, a) =
( -1 , +1 ) if m < 0
( 0 , +1 ) if m = 0 # canonical zero
( +1 , +1 ) if m > 0
- We use constants with
+1alignment by default for the outputs-1, 0, +1. - Alternative (manifest):
sgn_alignment = "carry_input"to output(±1, a)form ≠ 0.
Equivalences (with tie caveats):
H(m) = ( 1 + sgn(m) ) / 2
sgn(m) = 2*H(m) - 1
If H(0)=1/2 these equivalences hold pointwise; with other H(0) policies they hold away from m=0.
Heaviside step H
Classical (principal options at m=0):
H(m) = 0 if m < 0
1/2 if m = 0 # "half-maximum" convention (default here)
1 if m > 0
Symbolic lift (selector-carry):
H*(m, a) =
( 0 , +1 ) if m < 0
( 1/2 , +1 ) if m = 0 # default; see manifest
( 1 , +1 ) if m > 0
Manifest knobs:
heaviside_zero = "half" (default) | "right" (1) | "left" (0) | "alignment_sign"
If "alignment_sign", define H*(0,a) = (1, +1) if a >= 0, else (0, +1).
Ramp / ReLU
Classical:
ramp(m) = max(m, 0)
ReLU(m) = max(m, 0) # same function
Symbolic lift (selector-carry):
ramp*(m, a) =
( m , a ) if m > 0
( 0 , +1 ) if m <= 0 # tie handled by selector_tie; default prefers constant at m = 0
Equivalent forms (for magnitude):
ramp(m) = (m + |m|) / 2
ramp(m) = m * H(m)
Tie behavior follows your H(0) and selector_tie policies.
Clip (saturation)
Classical (clip(m; L, U) with L <= U):
clip(m) = min( max(m, L), U )
Symbolic lift (selector-carry):
clip*(m, a; L, U) =
( L , +1 ) if m < L # select lower bound constant
( m , a ) if L <= m <= U # pass-through
( U , +1 ) if m > U # select upper bound constant
Tie at boundaries (m = L or m = U):
Default selector_tie = "prefer_constant" keeps (L,+1) or (U,+1); declare otherwise if you want pass-through on equality.
Min/Max equivalences (magnitude channel)
For re-use in proofs and code:
min(m1, m2) = (m1 + m2 - |m1 - m2|) / 2
max(m1, m2) = (m1 + m2 + |m1 - m2|) / 2
clip(m; L, U) = min( max(m, L), U )
ramp(m) = max(m, 0)
Alignment follows the selected source (carry-from-source). For min/max of symbolic numerals (x1, a1), (x2, a2), select by comparing magnitudes and carry the winning alignment (a1 or a2); on ties, use the symbolic preorder (S_beta) or a stated selector_tie.
Collapse, order, and monotonicity
- Collapse:
phi( f*(m,a) ) = f(m)forf in {sgn, H, ramp, clip}given the chosenH(0)and tie policies. - Order:
rampandclipare monotone nondecreasing inm(for fixed bounds); selection by magnitude preserves order on each branch. - Stability semantics: Saturation (
clip) yields constants with+1alignment unless otherwise tagged—explicitly signaling that the output is bound-limited rather than input-driven.
Worked examples
A) Heaviside with default half-maximum at 0.
H*((-2.0, -0.3)) = (0 , +1)
H*(( 0.0, +0.9)) = (1/2, +1) # heaviside_zero = "half"
H*((+3.1, +0.2)) = (1 , +1)
B) Ramp with selector-carry and boundary tie.
ramp*((-0.1, -0.6)) = (0 , +1)
ramp*(( 0.0, +0.7)) = (0 , +1) # selector_tie = "prefer_constant"
ramp*(( 2.5, +0.4)) = (2.5 , +0.4)
C) Clip with input inside and outside bounds.
clip*((-5, 0.2); L=-1, U=2) = ( -1 , +1 ) # saturated at lower bound
clip*(( 0, 0.5); L=-1, U=2) = ( 0 , 0.5 ) # pass-through
clip*(( 5, 0.9); L=-1, U=2) = ( 2 , +1 ) # saturated at upper bound
D) Alternative tie via alignment_sign (manifest choice).
heaviside_zero = "alignment_sign"
H*((0, +0.3)) = (1, +1) ; H*((0, -0.8)) = (0, +1)
This makes threshold decisions auditably depend on the input alignment.
Implementation notes (manifest)
selector_alignment = "carry_from_source"(default) |"carry_input_always"|"constant_plus_one"selector_tie = "prefer_constant"(default) |"prefer_input"|"alignment_sign"heaviside_zero = "half"(default) |"right"|"left"|"alignment_sign"constant_alignment = +1unless explicitly tagged- Zero canonicalization: any output with magnitude
0→(0, +1) - Document numeric types for bounds (
L, U) and whether they are symbolic(L, a_L)(rare) or classical ((L, +1)default).
Takeaway
Piecewise and indicator functions behave as selectors: magnitude decides the branch, alignment comes from the chosen source. Clear tie policies make threshold behavior reproducible. Saturation exposes stability explicitly (constants report with +1), and all constructions collapse to the classical forms when a = +1.
Navigation
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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